Quantum mechanical reaction probabilities
via a discrete variable
representation-absorbing
boundary condition Green’s function
Tamar Seideman and William H. Miller
Department of Chemistry, University of California and Chemical SciencesDivision, Lawrence Berkeley
Laboratory, Berkeley, California 94720
(Received 5 March 1992; accepted 6 May 1992)
The use of a discrete variable representation (DVR) and absorbing boundary conditions
(ABC) to construct the outgoing Green’s function G(E+) =lim,,,(E+ie--H)-‘,
and its
subsequent use to determine the cumulative reaction probability for a chemical reaction, has
been extended beyond our previous work [J. Chem. Phys. 96, 4412 (1992)] in several significant ways. In particular, the present paper gives a more thorough derivation and analysis of
the DVR-ABC approach, shows how the same DVR-ABC Green’s function can be used to
obtain state-to-state (as well as cumulative) reaction probabilities, derives a DVR for the exact, multidimensional Watson Hamiltonian (referenced to a transition state), and presents
illustrative calculations for the three-dimensional H+H, reaction with zero total angular momentum.
I. INTRODUCTION
The canonical rate constant for a bimolecular chemical
reaction can be expressed as a Boltzmann average of the
cumulative reaction probability N(E) ,
k(T)=[2diQr(T)]-1~~a
dEe’E’krN(E),
(l.la)
which in turn is defined’ as the sum of state-to-state reaction probabilities (the square moduli of S-matrix elements) over all energetically allowed states of reactants
(denoted by quantum numbers n,) and products (denoted
by quantum numbers n,),
NW) = c I q,$W)
%“p
I 2.
(l.lb)
In Eqs. ( 1.1)) E is the total energy of the molecular system
and Q, is the reactant partition function. In some cases
(typically for unimolecular reactions) the microcanonical
rate constant is the quantity of interest and it is given in
terms of N(E) by
WE) = P77-@,W)
1-‘NW),
(l.lc)
where p,(E) denotes the density of reactant states per unit
energy.
Computation of the S matrix {S,,,)
in Eq. ( 1. lb),
however, requires a complete solution of the state-to-state
quantum reactive scattering problem, a level of detail (and
computational effort) which one would like to avoid if
interested only in the net rate constant. The major objective of this paper is thus a direct (but rigorous) evaluation
of the cumulative reaction probability, i.e., one which circumvents the necessity first to calculate the state-to-state
reaction probabilities and then to sum over asymptotic
quantum states.
Considerable theoretical effort has been devoted in recent years to the search for convenient, but nevertheless
rigorous methods for evaluating cumulative reaction probabilities and rate constants directly.2-13 Most of this effort
has focused on evaluation of the thermal rate constant
k( r)4g via the flux correlation analysis of Miller et al3 In
particular, we note the work of Park and Light6(c)76(e)*6(f)
who were, to the best of our knowledge, the only ones to
compute the thermal rate constant in full three dimensions.
Less attention has been devoted to the direct calculation of
the cumulative reaction probability N(E). lo-l3 References
10-13 (as well as the present work) have employed the
flux correlation-based expression33(b)
N(E) =$(2rrti)2Tr[F6(E-ZOFS(E--H)],
(1.2a)
where H is the total Hamiltonian of the molecular system
and F is a flux operator (vide infra). The microcanonical
density operator in Eq. ( 1.2a) is usually obtained from the
outgoing Green’s function (actually, the Green’s operator)
6(E-H)
= -iIm
G(E?),
(1.2b)
where
G(E+) = (E+-H)-l=lim(E+ie-H)-‘.
(1.2c)
E-t0
Equation ( 1.2a) for N(E) , although rigorously equivalent
to Eq. ( 1. lb), is clearly independent of any reference to
asymptotic (i.e., reactant and product) quantum states.
Significant progress in the search for an efficient approach to the direct calculation of N(E) was reported in
Ref. 13 (henceforth referred to as Paper I) based on a
discrete variable representation (DVR) 14,15of the Hamiltonian and flux operators in Eq. ( 1.2a), and the use of
absorbing boundary conditions (ABC) l&l7 to obtain a
well-behaved representation of the Green’s function in Eq.
( 1.2~). The desirable features of a DVR are well recognized. No integrals need be computed in constructing the
Hamiltonian matrix and the latter is extremely sparse, thus
allowing the efficient use of iterative methods for the linear
algebra (matrix inversion or diagonalization) involved.
DVR methods are widely used at present in essentially all
molecular problems which call for an L2 basis set. Due to
J. Chem. Phys. 97 (4), 15 August 1992
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62499-l 6$006.00
@ 1992 American Institute of Physics
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T. Seideman and W. H. Miller: Quantum-mechanical
2500
the difliculty of multidimensional numerical integration
(for evaluating matrix elements for polyatomic systems in
conventional basis sets), we believe that pointwise representations,” such as DVR, are the only way to make systematic progress in “ab initio chemical dynamics”; a DVR
would thus assume the role of the primitive Gaussian basis
of electronic structure theory.
Absorbing boundary conditions have been used by several groups in the past,‘6’*7 primarily in time-dependent
schemes. Goldberg and
wave packet propagation
Shore16(a) and Leforestier and Wyatt’6(b) employed absorbing boundaries in models of laser-induced dissociation,
and Kosloff et aI.17(a)s17(b)and Neuhauser and Baer
et al. 17(c)-17(h)have used ABC to study reactive scattering.
For our present application, the use of ABC provides a
simple means of imposing the outgoing wave boundary
conditions necessary for the proper and well-behaved representation of the Green’s function without having to include information regarding the asymptotic region (i.e.,
reactants and products), as is necessary, e.g., in a variational representation of G( E+ ) . lg ABCs provide, furthermore, a convenient means of limiting the DVR grid to the
interaction region, where the reaction outcome is determined.13 ,It is interesting to note that the use of an absorbing potential is closely related, and in many respects equivalent to the exterior scaling version of the complex rotation
method.20*21It appears to us, however, that the use of an
absorbing potential is considerably more convenient computationally than complex scaling of the coordinates. Application in Paper I to a one-dimensional potential barrier
and to the collinear H+H2 reaction showed the DVRABC approach to yield high accuracy while requiring a
modest computational effort and knowledge of the potential energy only in the interaction region.
The purpose of the present paper is to present a more
thorough derivation and analysis of the DVR-ABC approach of Paper I, to show how the same DVR-ABC
Green’s function can be used to obtain state-to-state Smatrix elements if these are of interest, to present a DVR
of the exact multidimensional Watson Hamiltonian,22’23
and finally, to illustrate the efficiency of the method by
application to the three-dimensional (J=O) H+H, reaction. The next section presents the derivation and analysis
of the DVR-ABC Green’s function, Sec. III derives the
DVR of the Watson Hamiltonian for a general multidimensional system, Sec. IV presents and discusses our
three-dimensional results, and the linal section concludes.
reaction probabilities
FIG. 1. A schematicillustrationof the physical(solid contours) and the
absorbing (dashed contours) potential energy surfaces for a generic A
+BC reaction in the collinear configuration. Q, and Q, are the normal
mode coordinates of the transition state.
tion region. Figure 1 illustrates schematically the physical
and the imaginary potential energy surfaces for a generic
reaction. The absorbing potential vanishes (or practically
so) in the interaction region, but becomes sizable outside
the physically relevant region of space. The considerations
in choosing the functional form of l?(q) and its location
were discussed in detail in Paper I [see also Ref. 17(c)],
namely, the absorbing potential is required to turn on
smoothly enough so as not to cause back reflection of flux
into the interaction region, but sufficiently rapidly to absorb the flux over as short a distance as possible (for the
sake of efficiency of the calculation). In Paper I, it was
pointed out that the absorbing potential I’(q) = 2E( q) may
be regarded as a coordinate-dependent modification of the
infinitesimal convergence factor in Eq. ( 1.2~). Below we
use e(q) and l?(q)/2 interchangeably to denote the imaginary part of the inverse Green’s function E+ -H. [Note
that in the conventional definition of G( E+ ), E is a positive
constant and the limit e--t0 must be taken at some stage.24
In the present approach, where E(q) is a positive function
of coordinates, the corresponding limit is that E(q) must
“turn on” sufficiently slowly at the edge of the absorbing
region and sufficiently far from the interaction region.]
Since E is a potential energy-like operator, its DVR is
diagonal. The DVR of the ABC Green’s function is thus
G(E+)=(EI+ie-H)-‘,
II. THE DVR-ABC GREEN’S FUNCTION
A. Cumulative
reaction
probabilities
This section is a direct follow up to Sec. II of Paper I,
the primary purpose of which is to present the derivation
of the note added in proof at the end of that section. We
thus suggest that the reader read this part of Paper I in
order to be familiar with the background.
As discussed in Paper I, an imaginary potential
--ir(q)/2
is added to the physical Hamiltonian, the purpose of which is to absorb flux that exits from the interac-
(2.1)
where H is the DVR of the Hamiltonian operator and E is
the diagonal matrix of the absorbing potential evaluated at
the grid points. For future use, we note that since
(El+ie-H)*G(E+)=I,
the real and imaginary parts of G=G,+iGi
following relations:
(2.2)
satisfy the
(H-EI).Gj=cG,
(2.3a)
(H-EI).G;(H-El)=--E--PG;E.
(2.3b)
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T. Seideman and W. H. Miller: Quantum-mechanical
reaction probabilities
2501
The flux operator F in Eq. ( 1.2) is defined most generally as
F=&Wf(q)lh
(2.4a)
where h is the step function
1, k-0
wr) =
i 0, if<0
(2.4b)
and f(q) defines, via the equation f(q) = 0, a dividing surwhich separates reactants from products.
<0
corresponds to reactants and f (q) > 0 corresponds to products. In most applications to date,3(b)-‘2 f(q) has been
chosen to be the reaction coordinate itself
=qF, in
which case, computation of the commutator in Eq. (2.2)
face
Employing Eqs. (2.3) and the cyclic permutation of the
operators in the trace, and noting that E and h commute
and that h*( I- h) =0 leads to the following remarkably
simple expression for N(E):
N(E) =Tr(r;G,;r,.G,+r;Grr~*Gj)
f(q)
f(q)
g&.5
F=kI:S(~F)
+~(QJ)
5 I,
(2.4~)
where PF is the momentum conjugate to the reaction coordinate, and it is assumed that H is of the form p2/2m
+ V(q) . In Paper I, though, the following form of the flux
operator was employed:
F=;
)I,
[T,Mf
(2Sa)
where T=p2/2m is the kinetic energy operator; this expression for F follows from Eq. (2.4a) since the potential
energy commutes with hlf(q)], both operators being functions only of coordinates. The DVR of Eq. (2Sa) is
F=;
[T,h],
(2Sb)
where h is the matrix representation of Eq. (2.4b).
In the present work, however, we employ two alternative forms, equivalent to Eq. (2.4a),
F=;
[H-E,h(f)]=-;
[N-&l---h(f)],
(2.6)
where use was made of the fact that the constant E commutes with h. The DVRs of Eqs. (2.6) are
F=;
(&h-h*@,
(2.7a)
or
F=-;
[&(I-h)-(I-h)&i],
(2.7b)
where
i&H--~1.
(2.7~)
Use of Eqs. (2.7) for the flux operator gives the following
DVR for the cumulative reaction probability:
=Tr(r,eGq,eG*),
where I’/2=c
(2.9a)
and
r,=(i-h)*r,
(2.9b)
rph*r.
(2.9c)
rr is the absorbing potential in the reactant valley and r, is
that in the product valley (see Fig. 1). Utilizing the fact
that I’ is a diagonal matrix and writing out Eq. (2.9a) in
terms of the DVR matrix elements gives
N(E) = F F rr(qj)rp(qy)
1G(qj,qj~;E+> 12,
(2-9d)
where {qj} are the DVR grid points and j denotes the
collection of grid point indices j = {j,j,,...,j,).
[In deriving
Eq. (2.9d), we utilized the fact that G is a complex symmetric matrix.]
Equation (2.9) has the desirable feature of being explicitly independent
the location
the dividing surface,
provided only that it lies somewhere between the reactant
and product absorbing regions. It is well known3(‘) that
the reaction rate, if evaluated exactly, is independent of the
choice of the dividing surface, but all expressions used to
date for the canonica13(b)-g and microcanonical’@i3 rate
constants contain an explicit choice of f(q) . The present
form manifestly shows this independence.
An important practical advantage of Eq. (2.9d) is that
it does not require the solution of a system of simultaneous
equations for Nt right-hand sides (Art being the number of
DVR points) as does Eq. (2.12) of Paper I. Since
G(qj,qj,;E+) need be known only for points qj and qjf
which lay in the reactant and product absorbing strips,
respectively, the linear system of equations
of
(El+k-H)gi=Ii
of
(2.10)
(where gj denotes the fib column of G, cgi)k=Gkj, and Ij
denotes the fib column of the unit matrix {li)k= ~k,j) need
be solved only for a small subset of vectors, in practice
15%-25% of the matrix order. This feature decreases the
computational effort involved considerably.
Equation (2.9) also lends itself to a simple qualitative
interpretation. The summand of Eq. (2.9d) can be thought
of as a transition probability from grid point qj in the reactant absorbing region to grid point qjp in the product
absorbing region. The cumulative result N(E) is then the
sum of the transition probabilities over all reactant and
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T. Seideman and W. H. Miller: Quantum-mechanical
2502
f
A’
V
E
\
e..
i\
4;.
47-4,
N(E)=
&‘LW-‘Jq’)
and the WKB approximation
dq J;
k(q%~~(
dq$$$g
and in the pre-exponential factor, the absorbing potential is
discarded
v(q)z
(2.13~)
&ixz.
The imaginary part of k(q”) is nonzero for q” <q;1 and
q” > q:, where r(q”)
is nonzero, as well as in the classically forbidden region qi< q” <q: With Eq. (2.13b), we
have
rrw )
(WKB) limit of N(E)
I G(q,q’;E+)
l+4E;;(;;,,)]},
(2.13b)
Im k(q”) S- 2fiv(q”)
In order to help develop insight into the primary result
of the previous subsection, namely Eq. (2.9), it is useful to
explore it for a one-dimensional potential barrier in the
WKB approximation, where the analysis can be carried
through analytically. Figure 2 illustrates schematically the
physical [V(q)] and imaginary [I’(q)] parts of the potential energy. We denote by q: and qf the left- and right-hand
turning points, respectively, and by qy and q; the corresponding points. where the absorbing potential is turned on.
The continuum limit of Eq. (2.9d) in the onedimensional case reads
dq j-s,
jTa
1
product grid points, in analogy with Eq. (l.lb) which is a
sum of reaction probabilities over asymptotic quantum
states of reactants and products. The practical advantages
of Eq. (2.9d) are that the reactant and product absorbing
strips lie immediately outside the interaction region, rather
than in the asymptotic regions, and that the sum is over
simple grid points, rather than over quantum states.
Finally, although Eqs. (2.9) have been derived utilizing special features of the DVR, the trace structure in Eq.
(2.9a) is clearly invariant to the choice of representation.
It thus holds for any basis set representation of the matrices involved in case for some applications a choice other
than a DVR might be preferred.
NV-9 = J:m
(2.12c)
+rp(d.
(2.13a)
” dq” Im k(q”) ,
Xexp -2
.[
I 4
where the integration ranges over q and q’ have been restricted to the regions where l?Jq) and l?Jq’) are nonzero
and Ghere v(q) =fik(q)/m.
@.&ing ailotied the absorbing potential to extend to
infinity, we are free to choose its magnitude to be small
throughout and make the standard semiclassical assumption l?(q) &E- V(q) for all q. The wave vector in the exponent of Eq. (2.13a) is thus approximated as
FIG. 2. A schematic illustration of the physical and the absorbing potential curves for a one-dimensional barrier problem (see Sec. II B).
B. The Wentzel-Kramers-Brillouin
-_
Thus
~..\.\.,~,.~~~,,;:
4;
_
._
r(4) =cw
_
d,
reaction probabilities
) 4”<4;1
rpw
=jygjq’ 4”>4F
=J$Gyzi, 41<4”<d.
(2.13d)
Using Eqs. (2.13d) in Eq. (2.13a) gives
12,
(2.11)
for the Green’s function is
(2.14)
where q< and q, are the smaller and larger, respectively,
of q and q’,
k(q) =
and
J
2m
x [E-v(q)
fir(q)/21
(2.12b)
where
-
_ -B(E)=
s’:dq”{;
4
[V(q”)-El.
The two factors in braces in Eq. (2.14) each give unity,
whereby
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T. Seideman and W. H. Miller: Quantum-mechanical
N(E) c&?-E),
(2.15)
the correct reaction (i.e., tunneling) probability for a onedimensional barrier in the WKB approximation.
In addition to verifying that N(E) reduces to the correct result in the WKB limit, the above analysis illustrates
several useful features of Eqs. (2.9). First, we note that the
result is independent of the location of the absorbing potentials, the only restriction being that qy <q; and q; > qf,
i.e., the imaginary potentials should vanish, or nearly so, in
the tunneling region, so as not to interfere with the reaction
dynamics. This region will be larger the lower the energy.
Conversely, for high energies, above the reaction barrier
8(E) 40, N(E) approaches the (correct) free particle result of unity, and qy and q; can be brought in arbitrarily
close to q=O. This is the limit of classical transition state
theory, which gives the classically exact result for N(E) in
one dimension. Quantum mechanics (e.g., the uncertainty
principle), however, does not allow the interaction region
to be reduced to a point, but perhaps even more important
is the fact that in multidimensions, the interaction region
must be sufficiently large to contain all the relevant reaction dynamics, including recrossing (transition
state
theory-violating)
dynamics that is necessary in order to
determine the correct net reactive flux.
C. State-to-state
S matrix
To conclude this section on the general theory of the
DVR-ABC Green’s function, it is interesting to show that
the same methodology can also be used in a very simple
way to obtain state-to-state S-matrix elements for a reaction. We alluded to this possibility briefly in the concluding
section of Paper I, but would here like to develop it further,
particularly in light of the algebraic simplifications of
G(E+ ) described in Sec. II A above.
We start with the formally exact expression for the
S-matrix
element quoted in Paper I
sf,i=$,i+i
[(@fIH--EI *i> + (Fig
[Xi)],
(2.16)
reaction probabilities
2503
tional coordinate. af is defined similarly, with R and Y
replaced by R’ = RABmc and r’ = rAs, respectively. Since pi
and @f are typically cut off before they enter the reaction
region, the Born term [the second term in Eq. (2.16)] is
also (rigorously) zero. With the above two observations,
Eq. (2.16) becomes
i
sf,i=z
(xfl G(E+ ) 1xi> -
(2.18)
Since Q>iand @f satisfy the correct asymptotic scattering boundary conditions [cf. Eq. (2.17c)], Xi and xf are
short-range L2 functions, so that the DVR-ABC representation of G(E+ ) is possible. The DVR of Eq. (2.18) thus
reads
S~,i=~~~.(H-EI)*G(E+)*(H-El)~~i,
(2.19)
where G (E+) is the DVR-ABC Green’s function of Eq.
(2.1), @i and @f are vectors in grid point space
(2.20a)
(@,i)j=Wy@i(qj),
(2.2Ob)
(af)j=wjl'2@ff(qj),
T in Eq. (2.19) denotes vector transpose, and {Wj} are the
(multidimensional)
DVR weights associated with {qj).
(The origin of a weight factor in the transformation to grid
point representation is illustrated in Appendix A for a specific choice of the DVR.) Using Eqs. (2.3) in Eq. (2.19)
gives the final result
i
S~,i=-~~~~~*G(E’)*E,.~i,
(2.21a)
where we used the fact that
-f ~+.~i=o
since pi and @f do not overlap, and replaced E by E, or ep
since pi and @f are nonzero only in the reactant and product absorbing regions, respectively. Since the absorbing potential E is diagonal, the explicit matrix expression corresponding to Eq. (2.21a) is
where
xi= (H-E)cPi
(2.17a)
x/=
(2.17b)
and
Shi(E)=-i
C ~~(qj)wjln,(qj)G(qj,qjr;E+)
ij’
XE(qjt)Wi{2@i(qjt)e
(&--E)Gp
pf,i is the amplitude of the i-f process contained in the
asymptotic form of the wave function pi, but since we are
considering a reactive process, as long as Cpi and @f are
nonreactive distorted waves, Eli will be zero. Specifically,
we chose @i and @f to be incoming waves (free or distorted) of the type used extensively in S-matrix Kohn variational reactive scattering calculations. ly For a collinear
A+BC system, e.g., pi behaves asymptotically as
Q>i(r,R) -~i(r)UI~1’2e-ik~,
(2.17~)
where di(r) is the initial vibrational state of BC, R is the
A-BC
translational coordinate, and Y is the B-C vibra-
(2.21b)
Equation (2.2 1) is a remarkably simple expression for
state-to-state S-matrix elements which parallels Eq. (2.9)
for the cumulative reaction probability. Thus, having calculated matrix elements G(qj,qj,;E’)
between grid points
in the reactant and product absorbing regions, one may
compute the cumulative reaction probability using Eq.
(2.9d) or individual S-matrix elements using Eq. (2.21b).
It should be noted that with the present formulation Of S,,
it is not necessary to “cut 08” the incoming wave functions
Cpiand @f in order to guarantee regularity at small R (as
it is for variational calculations”)
because the absorbing
potentials eliminate the small R contribution form Eqs.
(2.21).
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2504
T. Seideman and W. H. Miller: Quantum-mechanical
Whether or not the DVR-ABC approach will also
prove more efficient than the variational approach for the
study of discrete-discrete transitions remains to be tested
and is likely to be system and energy dependent. The
present method is, however, potentially of considerable advantage for the treatment of cases where dissociative channels are energetically allowed, including the case where the
final state of interest is a dissociative state. In the latter
case, Eqs. (2.16)-(2.21) all apply and the DVR-ABC procedure still generates the correct Green’s function. The
tinal state Qr is of the form
@faexp[ -ik(ql
sin cz+q2 cos cx)],
(2.22)
where k = ,/w,
and a labels the partitioning of the
translational energy among the fragments A, B, and C. The
treatment of such processes by the variational method
would require discretizing the continuum (of dissociative
states), which is considerably more involved.
Finally, it is useful to consider the one-dimensional
WKB limit of Eq. (2.21b) in analogy to the discussion in
Sec. II B. The continuum limit of Eq. (2.21b) reads
reaction probabilities
(2.26)
where the phase factors in G(E+ ) have been canceled by
the phase factors in ‘Pi and @f Again, each of the terms in
curly brackets in Eq. (2.26) integrates to give unity, yielding the correct WKB expression for the S matrix element
(2.27)
III. DVR OF THE WATSON HAMILTONIAN
In Paper I, we used a DVR grid in the normal mode
coordinates of the transition state in order to study the
collinear H+H, reaction. Although other coordinate systems can be equally efficient (see below), the transition
state normal mode coordinates provide a convenient choice
that makes it reasonably easy to obtain a grid appropriate
to the dynamics in the interaction region. It is also an
approach which can be readily generalized to more complex reactions. It should be emphasized that the use of
normal mode coordinates does not entail a normal mode
(i.e., harmonic) approximation for the Hamiltonian. The
kinetic energy in these coordinates is exact and the full
potential energy is used.
The normal mode Hamiltonian for a general polyatomic molecular system in three-dimensional space was
developed in a pair of elegant papers by Watson.22’23 Here
we show how a DVR of the Watson Hamiltonian can be
constructed in a convenient and computationally efficient
fashion, in contrast to the complexity of computing its
matrix elements in a conventional basis representation. The
Watson Hamiltonian was originally introduced22’23 and
subsequently employed25 for the treatment of bound state
problems, where the displacement vectors are measured
with respect to the potential minimum. The reference configuration can nevertheless be arbitrary, and for the present
purpose is chosen to be the transition state, i.e., the saddle
point of the potential energy surface.
dq’e~r(q)e~P(q’)~i(q)~~(q’)
SAi=-f J:, & S_m_
x(qlG(E+)149,
(2.23)
where we again refer to Fig. 2 and where e(q) = I’( q)/2 as
before. We use distorted incoming waves for Qi and @f,
$lr
@i(q)
= Jvo exp
(2.24a)
e4
@f(q)=
mexp
(2.24b)
where r], and 77~are the WKB phase shifts for the reactant
and
product
motion,
respectively,
an+
kdq)
,/2m[E- V( q>]/@. {Note that the absorbing potential E
i: associated with the convergence factor in G(E+ ) [Eq.
(1.2c)] and therefore does not appear in the wave functions. Note also that pi is a wave traveling to the right and
Qr a wave traveling to the left, the “incoming” direction in
each case.} Approximating
the Green’s function
(ql G(E+) I q’) as in Sec. II B, we find
-sq;
y dq”k,(q”)
s qr
4
-
: dq” - 4q”)
%v(q”)
s qr
1
’
A. Nonlinear transition
state, J=O
We first derive the DVR of the Watson Hamiltonian
for a general nonlinear transition state for zero total angular momentum. The generalization to J> 0 involves only
simple modifications and is given in Appendix B.
The dynamical variables of the Watson Hamiltonian
are the normal mode coordinates {Qk} and their conjugate
momentum operators
(2.25)
and with Eqs. (2.24) for the distorted waves, Eq. (2.23)
takes the form
where k= 1,...,F and F=3N-6,
N being the number of
J. Chem. Phys., Vol. 97, No. 4, 15 August 1992
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T. Seideman and W. H. Miller: Quantum-mechanical
atoms. In Eq. (3.1) and throughout this section, atomic
units (fi= 1) are used. Following Watson’s notation,22 Roman indices k and I refer to the F normal modes and bold
face quantities denote vectors in this space. The index
i=l,...,N
labels atoms and Greek indices a, /3, and y denote the three Cartesian coordinates x, y, and z of the N
atoms. Thus the composite index {iy) assumes 3N values
for i= l,...,N and y=x, y, and z. An overbar represents a
vector and two overbars a tensor, in three-dimensional
Cartesian space.
The vectors {$O’} are the atomic coordinates of the
reference geometry (e.g., the transition state), chosen so
that the center of mass is at the origin
2505
reaction probabilities
ij-=I;1%tQ>P,= 5 P&(Q),
(3.9b)
I=1
where
F
%Q> =
ksl
(3.9c)
gk,&
[Pl commutes with B,(Q) since the I= k term is absent
from the sum in Eq. (3.9a).] AV=AV(Q) is a quantum
correction to the potential energy surface V,22
AUQ) = -kJW8Q) 1
(3.10)
and V(Q) is the potential energy at atomic positions
{c(Q)}, given in terms of the normal mode coordinates by
(3.2)
[If Eq. (3.2) is not satisfied for the original Go’, one simply
redefines them by subtracting the center-of-mass position
= (8z lmi~o))/( XL rmi) from each atomic vector.]
R
Tirinertia
tensor of the reference geometry is
F.(Q)
=3”‘+
I
1
i=l
mi[~($o)-~o))
-
-z”)40)],
(3.3)
andthe F 3N-dimensional normal mode vectors, {Liy,k)
z={,?L,.~} are the eigenvectors of the mass-weighted force
constant matrix26
j,
(3.11)
Li,kQb
(3.12)
~=.i&k
Q
for jk=o, *l, *2 ,..., and k= 1,2,..., F. The potential energy operator is diagonal in a DVR14
(V+AV)j,jf=$,jf[ V(Qj) +Av(Qj>l,
(3.13a)
where
(&I WT. =r~pi,y,+)
IY
I
Next we introduce a grid in the normal mode coordinates {&}, e.g., the uniform grid used by us previously’3”5
N
To=
&.
(3.4)
SjJt
(3.13b)
’
=Sj,,j~i;sj2J~a
.Sj&
r’
which correspond to nonzero eigenvalues. The Coriolis
coupling constants are defined as
The matrix of the zeroth-order kinetic energy, the first
term in Eq. (3.7), is also rather simple
N
~,&J=
c
&kx
x&l=
(3.14a)
(3.5)
-Fl,k
i=l
and the centrifugal distortion constants as
‘i;k= ~
Jlni 5(~“.Li,k)
(3.14b)
(3.6)
In terms of the above quantities, the complete molecular Hamiltonian
for J=O is given as
and the one-dimensional Cartesian kinetic energy matrices
are given asi3*15
H=
( ‘Li,k~‘)’ + ‘~“L,k’)
1
.
i=l
-~
where
1
F 1
2 -~+gTG+v+av,
(3.7)
I=12
where F=F(Q)
inertia
is the inverse of an effective moment of
F(Q) = [To+‘cQ>
l-‘-‘b-
[To+TtQ)
I-’
(3.8a)
with
(p2), =(-lp-/;
I J*Ji
AQf
2 ~a~a,p(Q)=~
GB
(3.8b)
= 5
F is the vibrational angular momentum
12/(jl-ji>2,
h=ji
L5%
(3.14c)
The rotation-vibration coupling term, the second term
in Eq. (3.7), is somewhat more complicated. Watson22 has
shown that the order of the ?i and the p factors is unimportant, i.e.,
+p.&
‘7;<Q)= kil Q&c.
213,
~aq+a,p(Q),
(3.15a)
(3.15b)
F
ii=
c
k;l=l
Bk,l&h
which may be expressed conveniently as
(3.9a)
=
~~a,dQhwp
(3.15c)
Using Eq. (3.15b), this term becomes
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T. Seideman and W. H. Miller: Quantum-mechanical
2506
Z-j==
5 ; ;, ~,~kP~?,,lQk’Pll~~,~(Q)
9 > ,
(3.16)
= aCp; B~(Q)P~~~~(Q)~,~(Q),
2 ,
so that its DVR is
grid representation allows physically inappropriate regions
[although one may experisimply to be discarded 13~14(d)~15
ence slow convergence if the region defined by Eq. (3.20)
is energetically accessible2’].
B. Linear transition
Use of Eq. (3.15c), on the other hand, gives the DVR
matrix element as
(+‘jFa’?r)jJt= z %(Qj>$XQj>*%ttQjt)
reaction probabilities
(PQ,,)j~ts
(3.17b)
the only difference between Eqs. (3.17a) and (3.17b) being
the grid point at which p(Q) is evaluated. In order to
obtain a manifestly symmetric matrix, we thus take the
average of Eqs. (3.17), which gives
state J=O
The analysis for a
complicated than that
resulting Hamiltonian
(3.7) vanishes in the
quantity
linear reference geometry23 is more
for a nonlinear geometry,22 but the
is simpler. The term AV in Eq.
linear case, and y(Q) is a scalar
IO
P(Q)=~~~+~~Q~~~,
(3.21)
where IO is the moment of inertia of the (linear) reference
geometry. Choosing the reference configuration to lie along
the z axis,
N
IO= 2 m,(r~‘,2,
i=l
(3.22)
and
(f*,FF)j,j,=~ z ~dQj>*l$IQj>+FtQjr) l*~~~(Qj~)
b(Q)=
(3.18)
X (P[Plt)j,jt*
Qkbk
(3.23a)
bk= ig, $$$‘Liz,k*
(3.23b)
k;,
with
The matrix of the momentum operators in Eq. (3.18) is
(PPP)j,j*=(@)jjp
for I= I’ and thus given by Eq. (3.14). For l#I’,
The molecular Hamiltonian is then
(3.19a)
CpPP)jjt =s~~~(pl)jlj;(plf)j~,j’~ P
where
(3.19b)
$$= ,=,fi+,,
3 , sjkdL
and
0,
A=.$
C&--j;)-‘,
j&l’.
(3.19c)
Except for Rqs. (3.14~) and (3.19c), which give the momentum matrix elements for a specific DVR, all the above
formulas apply equally to any choice of the DVR.
The discrete variable representation of the Watson
Hamiltonian is thus quite simple to construct, considerably
simpler than the correspondinn basis-set representation.25
In particular, the factor &+b(Q)]-‘,
which in the latter
class of representations needs to be expanded in a power
series, assumes a simple form in the DVR. The major difficulty in evaluatkg gq. (3.8a) in a basis set is that regions
of space where [l,+b(Q)]-’
is singular, i.e., points Q at
which
det[Tc+z(Q)]
=0
(3.20)
may be encountered. This difficulty is, however, straightforward to deal with within a DVR, since in contrast to
basis set representations which explore the entire space, the
H=
7 &L(Q)
(T%$)
i- V(Q),
(3.24)
where rX and rY are the x and y components of the vibrational angular momentum vector [Eq. (3.9a)], respectively.
As was pointed out by Watson,23 the only eigenvalues
of H of interest for the linear geometry are those corresponding to the zero eigenvalue of (J,-n-J
(of 7rZfor J
=O). These are constructed, as shown below, by forming
appropriate combinations of the degenerate bending
modes.
C. Example-the
H + H2 reaction
In this subsection we describe explicitly the form of the
Watson Hamiltonian for the H3 transition state. The four
normal mode coordinates for H, are shown in Fig. 3. Qi
and Q2 denote the symmetric and antisymmetric stretch
coordinates, respectively, and QS and Q4 are the two degenerate bending vibrations. The normal mode coordinates
are given in terms of the Cartesian atomic coordinates as
d-&-Q,=-q,+w-29,
(3.25a)
(3.25b)
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T. Seideman and W. H. Miller: Quantum-mechanical
__.
.=j
-
.
-
-LI
-
Q,
reaction probabilities
2507
: .-. I - ..~
-The standard-procedure to ensure~%hat 7rZ=0 is ‘tz’
transform from the two Cartesian vibrational coordinates
Q3 and Q4 for the degenerate bending modes to polar coordinates,
(3.28a)
p= &zz
-
e--7
.-
Q2
#=tan-’
f
t
0
e
t
a
@
Q4
2
(
.
(3.28b)
1
In terms of the latter coordinates
(3.30)
X
.._
f
_--. --3
~~z
>a
.
c -7~1
FIG. 3. The normal mode coordinates of the three-dimensional H+Hz
transition state. Q, and Q2 are the symmetric and asymmetric stretch
vibrations, respectively, and Q, and a are the degenerate bending vibrations.
and the requirement that the associated eigenvalue be zero
is,.equivalenr to ,omitting the 4 degree of freedom. The
complete Hamiltonian for the J=O H+H, reaction is thus
a function of three coordinates (Ql,Q2,p),
H=f #+@+$I +2(b$+n”Y>
+ V(QI,QBP)
(3.31a)
and the corresponding volume element is
(For reference, we note that the three inter-particle distances are given in terms of (Q,,Q2,p> by
dA Q4=rlv-2r2y+r3p
(3.25d)
l/2
r12=
where mu is the mass of a H atom and rt denotes the H-H
bond length at the transition state. The nonzero elements
of the inverse transformation (3.11) are
J;,
J&,2=-
J&,1=
-
(3.31b)
dr =pdpdQldQ2
(3.25~)
ip2+i
(
1
1
K’
l/2
gp2+i
(
1
&+Q1-fiQ2>’
&+Q1+fiQ212
1
1 Tz’
8 and fi remain Cartesian-like kinetic energy operators
;P&$,
La,3=L2y,4=
(3.26)
-2
and the moment of inertia [Eq. (3.22)] is
(3.27)
The centrifugal distortion constants of Eq. (3.23b) are
readily obtained using Eqs. (3.26)
bl= &,
(3.28a)
b2=b3=b4=0,
(3.28b)
and the vibrational angular momentum
using Eq. (3.9a), as
(3.32)
an aopropriate DVR for which is given in Sec. II A. For
the p degree of freedom, the kinetic energy operator is
6’
Io=2mH(&)2.
-,&1,2,
k
(3.33)
a DVR for which is is derived in Appendix A.
The vibrational angular momenta in Eq. (3.31a) are
d+4=-[p’$+Qi
vector is given,
rx= (Q.&k-Q&d,
(3.28a)
~y=(QG’rQ&),
(3.28b)
n;= (Qd’4-Qd’d.
(3.28~)
($+$$)+l-(l+p$)
(3.34)
where the factors
(3.35a)
and
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Chem. Phys., Vol.
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4, 15
Augustor 1992
T. Seideman and W. H. Miller: Quantum-mechanical
2508
reaction probabilities
Eq. (3.35b) is given, using Eq. (3.19c), as
(3.3;b)
(1+2Q2&)
have been grouped as such since each of these operators is
antisymmetric in its respective one-dimensional space, and
thus the Hamiltonian is manifestly symmetric in the full
three-dimensional space. The DVR of the operator in Eq.
(3.35a) is given in Appendix A and that of the operator in
~~~~~~~~~~~
= i [ (fi)j,~;Sjzj;6ip,j~ + (e)j2,jlSj,,jiSjdi
+
( 1+2Q2&)jJ,={~Al)j-Y(j+j~)/(jkf),
;ii’.
(3.36)
To summarize, therefore, the DVR of the J=O Watson
Hamiltonian for H3 is
+ (e)jp,jiSj,,j;Sj2,ji I +k ( &+
( 1+2Q 2 T&J
(Q-12)2Sj~,j~(P2p)j~j~-Sj,~~Sj~~~Sj~,j~+
Q 7 1-2Sjl,j;piSjpjLC@)j2Ji
( 1+p 2)
1 j& \
1 +Sj,j;Sj~j;sjj~j~V(Q:yQ~yPj~~
/ i&j J
(3.37)
I
where
Q :’=.idQl,
Q t =hAQ2,
and pip are defined in Appendix A. The DVRs of the various operators in Eq. (3.37) are given in Eqs. (3.14),
(ARC), (3.36), and (A9c).
absorbing potential to be most efficient when placed along
the reaction coordinate, i.e.,
r(Q)=r(lQ21).
(4.1)
For the functional form of I?(q) , we used in most examples
the Woods-Saxon form
2a
r(4) =
IV. RESULTS AND DISCUSSION
In this section, we present and discuss the cumulative
reaction probability for the three-dimensional (J=O) H
+H, exchange reaction computed using the method described in Sec. II A. Most of the examples shown below
employ the normal mode coordinates Hamiltonian developed in Sets. III B and III C. Similar calculations were
performed also using Radau coordinates,28 defined in Appendix C, and the efficiency of the two coordinate systems
is compared briefly.
A primitive DVR grid is first laid down along the normal mode coordinates of the transition state {Ql,Q2,p) for
-QI,~&QIGQI,,,, -Q2,,,<Q6Qzm,, and P = pjPy
jp= l,...,Nb, where Ql,max and Q,,,, define cut-off parameters along the vibrational and the translational coordinates, respectively, and Nb is the number of GaussLaguerre abscissas (Appendix A). As in our collinear
study, the primitive grid is next truncated by defining an
energy cut-off parameter V, and discarding all grid points
{Q,,,Q2,,pjp3at which V(Ql,j,,Q2,j2’pjp)
> V~‘3p14(d)*15
The discarded points are in classically forbidden regions
for states with energies below V, and thus superfluous for
the description of states with energies sufficiently smaller
than V,. The energy and radial cut-off parameters are determined, as shown below, by carrying out the appropriate
convergence tests.
The optimal location of the absorbing potential and its
functional form were discussed in detail in Paper I. In
accord with the conclusions obtained there, we found the
l+ekdGh4-q)/~l
*
As discussed in Sec. II A, the computational effort involved in evaluating Eq. (2.9d) depends on both the total
number of grid points Nt and the number of points which
lie in the reactant absorbing region Nr; The former number is the order of the Hamiltonian matrix in Eqs. (2.7)(2.9) and the latter is the number of right-hand sides in
Eq. (2.10). In order to reduce Nr, to the essential minimum, l?(q) in Eq. (4.2) was set to identically zero for
small values of q, where its value is otherwise exponentially
small. Calculations using a power law turn on for the absorbing potential produced similarly accurate results, but
required a considerably larger value of Q2,,, (and consequently a larger number of grid points), this being in accord with earlier conclusions based on one- and twodimensional calculations.i3
In all calculations shown below, we used the physical
bend frequency of the H3 transition state ~~~0.004 a.u. for
the frequency parameter w introduced in Appendix A. The
results were found, however, to be insensitive to the value
of o over a wide range, 0.003<w<O.O12 a.u. For values of
o outside this range, a considerably larger number of grid
points (NJ was required.
Figure 4 shows the three-dimensional cumulative
H+H2 reaction probability as a function of the total energy. We used the double many-body expansion (DMBE)
potential energy surface of Varandas and co-workers.2g
The circles show the reaction probabilities of Chatfield et
3o who employed the same potential energy surface and
al.
J. Chem. Phys., subject
Vol. 97, toNo.
4,15
August
1992
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T. Seideman and W. H. Miller: Quantum-mechanical
I’
*
t
”
a
u
*
1167
( 174)
16.0
‘I
-1
1
151
301
318
20.0
40.0
60.0
I
10.0 -
12.0
8
%
g
3
E
2509
reaction probabilities
6.0 -
8.0
263
( 65)
0.0'
0.5
'
g
0.7
*
0.9
2.0 -
t
1.1
1.3
'
1.5
'
,.
-2.0 ’
\-,-. 0.0
Qzmax
I
_. .~_
80.0 _ .-- --
E (ev)
FIG. 4. The cumulative reaction probability for the three-dimensional
(J=O) H+Hr reaction using the DMBE (Ref. 29) potential energy surface. The circles show the results of Ref. 30. Indicated above the curve are
the total number of grid points employed in the calculation and in brackets are the number of points which lie in the reactant absorbing region as
a function of the energy.
computed N(E) by solving for the state-to-state dynamics
and using Bq. ( 1. lb). Above the different energy regions of
the probability curve are indicated the total number of grid
points which were required in the calculation and the corresponding number of points in the reactant absorbing region. At low energies, Ego.75 eV, where the reaction proceeds nearly exclusively via collinear configurations, only
small values of p need be considered and the number of
points required in order to describe the bend motion accurately is as small as Nb=3. In addition, a relatively low
energy cutoff suffices V,= 1.5 eV, and the density of points
in the translational and vibrational degrees of freedom
(which is determined by the deBroglie wavelength at energy E) is relatively low. As a result, a rather small number of grid points N,=: 170 is sufficient in order to generate
the probability curve. As the energy is increased, the interaction region includes a larger range of bend angles and the
number of points required in order to describe the bending
vibration accurately increases. We found that four GaussLaguerre quadrature points are required at 0.75 <E < 0.95
eV,fiveat0.95<E<l.l5eV,andsixat
1.15<E<1.55eV.
The gradual increase in Nt and Nr, in Fig. 4 also reflects
the increase of the energy cutoff and the density of points
with total energy. It is remarkable, however, that even at
the highest energy in Fig. 4, a relatively small number of
grid points (N,z 1000-1160) is adequate. It should be
noted that the number of points which lie in the reactant
absorbing region &?li,,Q2j,,pjP}d’,
is only a small fraction
of the total number of grid points, which results in significant computational savings.
The rough accuracy (2%-3%)
probability curve
shown in Fig. 4 was generated using a relatively narrow
grid in the reaction coordinate direction &,,~30-40,
and a rapid turn on of the absorbing potential, q=: l-2
in Eq. (4.2) (distances are measured in units of
FIG. 5. Percent error in the cumulative reaction probability vs the radial
cut-off parameter Q,,,., (measured in units of mass”*>< bohr). N(E) is
computed using transttion state normal mode coordinates (see Sec. III C)
(-) E=0.8 eV, (-*-.-.)
E=l.Ol eV; (---) E=1.49 eV. The lower and
upper rows of abscissas at the top of the figure indicate, respectively, the
total number of grid points employed in the intermediate energy calculation, and the corresponding number of points in the reactant absorbing
region.
mass1’2 X bohr) . Arbitrary
accuracy in the cumulative
probability can be obtained with a larger computational
effort. As shown below, this requires a smoother turn on of
the absorbing boundary (i.e., a larger value of 7) and consequently a larger Q,,,,. In Big. 5, we examine the convergence rate of the cumulative reaction probability with
respect to the radial cut-off parameter Qz,,= The ordinate
shows the percent error in N(E),
,““Cf(E)
-p”-fiCal(E)
Wxact(E)
x 1oo
(4.3)
with respect to the fully converged reaction probabilities of
Zhang.31 In order to compare with the latter results, all
calculations were carried out using the Liu-SiegbahnTruhlar-Horowitz
potential energy surface.32 The three
curves of Fig. 5 correspond to three widely spaced energies.
The lowest total energy E=0.8 eV is slightly above the
threshold of the second (u= 1) vibrational channel and a
total of six rotational channels are open. At the intermediate energy E= 1.01 eV, a total of eight rotational channels are open, and at the highest energy considered E
= 1.49 eV, three vibrational and 15 rotational channels
contribute to the cumulative reaction probability. The
lower and upper rows of abscissas shown at the top of Fig.
5 indicate, respectively, the total number of DVR grid
points retained in the intermediate energy (E= 1.01 eV>
calculation and the corresponding number of points in the
reactant absorbing strip. As shown in Fig. 5, the error falls
-42-50 with the low energy curve
below 0.1% at Q2,maxconverging somewhat earlier than the two higher energy
curves. [The width of the grid, determined by Q,,,,,, depends on the translational, rather than the total energy.
Studies of the convergence properties of N(E) for a onedimensional model13 have shown that Q,,,,, increases as
J. Chem. Phys., Vol. 97, No. 4, 15 August 1992
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_
-~
2510
T. Seideman and W. H. Miller: Quantum-mechanical
--1%
.- .-x26
10.55
750
236
reaction probabilities
189
7
373
393
1180
\
10.0
ip?‘:
g
6.0
k2
,:
.I
.:
’
2.0
1
‘L.,
0.5
1.5
2.5
3.5
-11.0 L
1.0
,’
/’
,’
,’
I’
,’
,
,_--
*- ,’
,’
I
1.6
2.2
2.8
3.4
Qr.,,,(a.4
Vc (ev)
FIG. 6. The same as in Fig. 5 vs the energy cut-off parameter V,
the translational energy is decreased, in accord with the
qualitative discussion of Sec. II B. Since at each of the
three total energies examined several rovibrational channels are open, their convergence behavior is similar.]
Figure 6 shows the percent error in the cumulative
probability as a function of the energy cut-off parameter
V, again at three different energies. The total number of
grid points and the number of points in rr at E= 1.01 eV
are indicated above the plot as in Fig. 5. As expected, a
larger energy cutoff is required in order to attain a given
degree of convergence the higher the total energy. Thus, an
accuracy of 0.1% is obtained with Vc--,2 eV at a total
energy of E=0.8 eV, with V,z2.6 eV at E= 1.01 eV and
with Vc=:3.3 eV when the total energy is increased to 1.49
eV.
Similar results were obtained with the Radau coordinate system described in Appendix C. Again, low accuracy
results were attained with three to five grid points in the
bend degree of freedom (y in Fig. 10) and a total of
~280-1300 grid points depending, as in Fig. 4, on the total
energy. Figures 7 and 8 illustrate the convergence rate of
the cumulative probability with respect to the radial cut-off
parameter along the reaction-type coordinate Q,.,,, defined in Appendix C, and with respect to the energy cut-off
parameter V, respectively. Comparison of Figs. 5 and 6
with Figs. 7 and 8 indicates that the efficiency of the two
systems of coordinates for the reaction considered here is
comparable. Both sets of coordinates have the advantage
that the angular variable assumes (for a collinear transition state) only a relatively narrow range of values. An
advantage of the Radau coordinates is that the kinetic energy operator [Eq. (C2)] (and therefore its DVR) has a
simple form which involves no first or mixed derivatives.
The more complicated structure of the kinetic energy in
the normal mode coordinates [Eq. (3.31a)] results in a
certain lost of sparsity of the corresponding DVR matrix.
Nevertheless, while the former system of coordinates is
suitable for triatomic systems with a collinear transition
state, the latter is completely general and, since all degrees
of freedom are defined as deviations with respect to the
FIG. 7. The same as in Fig. 5 using Radau coordinates (see Appendix
a.
reference geometry, is expected to be as efficient for the
study of an arbitrary configuration of the transition state,
as well as for higher-dimensional chemical reactions.
Finally, in Fig. 9, we examine the role played by the
vibrational angular momentum coupling terms and by
nonseparability and anharmonicity of the bend motion,
two effects which are often neglected in approximate studies. The solid curve of Fig. 9 shows the exact reaction
probability, computed using the full Hamiltonian of Eq.
(3.37). The dashed curve illustrates the effect of neglecting
vibrational angular momentum [the second term in Eq.
(3.31a)] and the dotted-dashed curve takes the approximation one step further and replaces the exact potential by
a separable, harmonic approximation to the bend motion
+
+b$p2.
(4.4)
As can be readily shown, the latter approximation is equivalent to replacing the three-dimensional cumulative probability by a sum of energy-shifted collinear-like reaction
probabilities
UQl,Qzg)
t
V(Ql,Qz,p=O)
241
757
387
1395
331
1113
1.0
3.0
4.0
Vc (e-V)
FIG. 8. The same as in Fig. 6 using Radau coordinates.
J. Chem. Phys., Vol. 97, No. 4,15 August 1992
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T. Seideman and W. H. Miller: Quantum-mechanical
18.0
6.0
i
:
:
16.0
:
!
14.0
!
//
:
:
12.Q
4
1.55
10.0
E WI
FIG. 9. The cumulative reaction probability for the H+H, reaction (-)
using the exact Hamiltonian [Eq. (3.37)]; (---) neglecting vibrational
angular momentum; (-a-*-*) within the separable harmonic approximation of the bend motion [Eq. (4.5)].
(4.5)
an approximate approach which has often been used by
Bowman and co-workers33 in the framework of a hierarchy
of reduced dimensionality theories. As shown in Fig. 9, the
neglect of vibrational angular momentum leads to relatively minor changes in the cumulative reaction probability
throughout the energy range considered, although its effect
becomes more pronounced as the energy is increased. Anharmonicity, and more importantly nonseparability, of the
bend motion is seen to play a more important role in the
dynamics.
V. CONCLUSIONS
This paper has extended our previous work on the
direct evaluation of quantum mechanical cumulative reaction probabilities in several significant ways. Using the
DVR-ABC Green’s function, Sec. II first derived a form
for the cumulative reaction probability (2.9) that is convenient both conceptually and computationally. One particularly pleasing feature of Eqs. (2.9) is that N(E) is
explicitly independent of the location of the dividing surface, a general result which has been known to hold for an
exact theory, 3(a) but which was not manifest in previous
expressions for the canonica13(b)-g or microcanonical’0-‘3
rate constants. It was also shown that the DVR-ABC
Green’s function can also provide a simple route to stateto-state S-matrix
elements for reactive systems, which in
principle remains useful even in cases where break-up (i.e.,
dissociative) channels are open, including the case where
the final state of interest is dissociative.
In Sec. III, the Watson Hamiltonian,22’23 a general and
exact molecular Hamiltonian originally introduced for the
description of bound state problems, was shown to assume
a relatively simple and computationally convenient form
when expressed in a DVR, in contrast to the difficulty of
computing its matrix elements in a conventional basis set.
reaction probabilities
2511
The DVR of Watson’s Hamiltonian was used in the
present work for scattering calculations, where a natural
choice for the reference configuration is the transition
state. The form derived in Sec. III is, however, general, and
is expected to be of utility also for bound state problems, in
which case, the reference configuration would be the potential minimum.
Application in Sec. IV to the three-dimensional (J= 0)
H+H, exchange reaction fully verified our expectations,
based on previous collinear results, that the DVR-ABC
approach would render the direct calculation of N( E) considerably more efficient than the corresponding state-tostate calcul?tion. The results presented in Sec. IV suggest
that a major computational savings is afforded by our ability to restrict the DVR grid to the interaction region,
where the important dynamics takes place. Since the reduction in the required number of grid points as compared
calculation manifests itself in each of the
to a state-to-state
degrees of freedom, the computational advantage of the
direct calculation increases dramatically as the dimensionality is increased. It was also seen for this example that
neglect of vibrational angular momentum terms in the
Hamiltonian causes essentially no error, particularly at
lower energies. This and other such standard approximations will be helpful (perhaps essential) in carrying out
calculations for more complex reactions.
Finally, a very encouraging observation of the present
study is that degrees of freedom which are more passively
coupled to the reaction dynamics require fewer grid points
than those more intimately involved in the reaction. For
the H3 system, e.g., the symmetric and antisymmetric
stretch modes Ql and Q2 are the two most actively involved
in the bond-breaking and bond-making processes, with the
latter (the reaction coordinate) requiring considerably
more grid points than the former. The bending mode p is
more of a spectator and only three to five grid points are
necessary in order to characterize it accurately. (Assuming
that the bend motion is completely uncoupled, however,
was seen to cause noticeable error.) One would like to
believe that this behavior will be typical in more complex
reactions, namely that the (many) relatively passive degrees of freedom will require comparatively small DVR
grids for an accurate description. Verifing this expectation
will be one of the interesting avenues for future work.
ACKNOWLEDGMENTS
We would like to thank Dr. J. Z. H. Zhang for providing us with the H + H2 reaction probabilities. This work
was supported by the Director, Office of Energy Research,
Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contact No.
DE-AC03-76SFOOO98. T.S. is supported by the National
Science Foundation, Grant No. CHE-8920690.
APPENDIX A: GAUSS-LAGUERRE DVR FOR TWODIMENSIONAL DEGENERATE OSCILLATOR
A bending vibration about a linear reference geometry
(a potential minimum or a saddle point) is doubly degen-
J. Chem. Phys., Vol. 97, No. 4, 15 August 1992
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2512
T. Seideman and W. H. Miller: Quantum-mechanical
erate. It is thus useful to define a DVR for the bending
degree of freedom based on the eigenfunctions of a twodimensional degenerate harmonic oscillator. In order to
construct eigenfunctions of the total angular momentum, it
is convenient to transform from two-dimensional Cartesian
coordinates to polar coordinates. The Hamiltonian for a
two-dimensional degenerate oscillator in Cartesian coordinates - CO< ql,q2 < CO,
(Al)
Gauss-Laguerre quadrature, and (Wj] are the corresponding weights.34 The grid points in p space are then
pi’
2-+-+-2.
pap
i a2
2
p a4
WI
where
(A7d)
o=-;($+i$),
on,i+=;
[(2n+1)6,,~+ns,-1,,,+n’6,+l,~,]
&FJNg’
(2?Z+l)L,(Xj)L,(Xjt)
n=O
+n[Ln(xj)Ln-l(xjf)
and
The &dependent portion of the eigenfunctions of Eq. (A2)
is e”@and we consider here the case I=0 corresponding to
zero vibrational angular momentum. The radial eigenvalue
equation in mass weighted space using atomic units (m =+i
=l) is thus
(A8b)
and thus
(A34
Wb)
(A84
e.g., the basis set representation is
OjJt=T
P=&fTZ
2.
J
For the operator
is thus replaced by the following in polar coordinates:
a2 ia
reaction probabilities
+Ln-l(xj)Lfl(Xjt)
19
(A8c)
where Nb denotes the number of grid points. The second
operator required in Sec. III is
0=1+pa
NW
aP ’
whose matrix in the basis
On,nt=n&-l,nI
{I),)
is
-n’Sn+l,nt.
(Agb)
The grid representation of Eq. (A9a) is then
[ -;
(~+~~)+~~2P2--E.]~~(p)=o,
(A4)
for which the eigenfunctions (normalized with respect to
dr=pdp) are
*n(p) = I/GJe-~p2’2L,(op2),
(A5)
{L,} being Laguerre polynomials,34 and the corresponding
eigenvalues are
E,=w(2n+l),
n=O,l,*.. ‘.
Nb-1
Oj,$= Jwiwi
-Ln-l(Xj)Ln(xjr)]-
%n~=(~nlollcst~>~
(A74
%I= (&zlQ),
(A101
then
vn=
s 0
=
s0
CoP dP hAp>Wp)
O”p dp 6
its matrix elements in the DVR are given by
Oij,=<j[
=
Olj,)
s
om dx eex
~,(~p~)a(p)
exp
8’2
L,(~Prp(~)l,
(All)
-z
where x=wp2. With a Gauss-Laguerre quadrature for the
integral,
Un=2 w$n(Xj)
i.e.,
ODm=UOUT,
WC)
Finally, we note the DVR of a Hilbert space vector. If
u, denotes the representation of the state I@) in the basis
(A5),
C-46)
A grid point representation for an arbitrary operator 0
may now be constructed following the standard DVR
methodology. l4 Denoting by O,,+I the matrix elements of 0
in the basis (A5 ) ,
2 n[Ln(Xj)L,-l(Xjf)
n=O
(A7c)
where Ujn = JwjL,(Xj) are the elements of an orthogonal
transformation between the grid and the basis representations, {Xi> are the quadrature abscissas associated with
=;
i
u-Jn-(
e+
-J20 @>[PCxj)1
w@ “2
20 ) a[[p(xj)l*
(A121
Since U defines the transformation between the basis set
and the grid representations (ARC), the DVR of the vector
1(P) is identified as
J. Chem. Phys., Vol. 97, No. 4.15 August 1992
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T. Seideman and W. H. Miller: Quantum-mechanical
wp l/=
JJ,=
aq[p(Xj)]
J
( 2cd 1
reaction probabilities
2513
(A13)
=wy=@(pj).
[The multidimensional weight Wj of Eqs. (2.20) and (2.21)
takes into account the quadrature weights in all degrees of
freedom.]
As an aside, we note that the Gauss-Laguerre abscissas are given approximately by
(A14a)
Xjz [ (i-f)~]~/(4Nb+2),
so that the grid points cpi> are approximately
(j-$h/
p/z
(A14b)
/WiTi%.
To a rough approximation
spaced with spacing
then, the bj>
grid is equally
I
.,_~
(A14c)
Ap=-Jyz&T2
FIG.
10. The triatomic
Radau
coordinates
(see Appendix
C).
Equation (A14c) is a useful relation to bear in mind when
choosing a reference frequency w and a number of grid
points iV&
APPENDIX B: DVR OF THE WATSON HAMILTONIAN
-
(J+K+ 1) (J--K)&,K~-II,
(B3bj
J>O
and
In addition to the F = 3N- 6 degrees of freedom in the
J=O Hamiltonian of Sec. II a, for total angular momentum
J> 0, there is an additional degree of freedom that participates in the molecular dynamics, namely the component
of the total angular momentum along a body-fixed axis
(typically chosen to be the z axis). The magnitude of the
total angular momentum, quantum number J, is of course
conserved.
The Watson Hamiltonian for a nonlinear reference geometry and J>O is==
H=
F 1
c, -ff+;
I=12
&=o+;~~
(5rt-iT).~(~+Z)+V+AV,
(Bl)
--
J+ J.F$,
W)
where Z and p are discussed in Sec. II a and 7~ (J,,J,,,J,)
is the negative of the total angular momentum operator,
the sign reversal having been introduced by Van Vleck35 so
that the commutation relations (and matrix elements) of
the operators Jp J, and J, would be the conventional ones.
Since the potential energy surface V and the correction
term A V do not involve the angular momentum degrees of
freedom, the standard angular momentum representation
can be employed conveniently for that degree of freedom.
We thus use a DVR for the F vibrational-like degrees of
freedom and the conventional angular momentum basis
1JK) for the rotations. The matrix elements of J in this
basis are
(J~)K,P=; I: &J--K+U(J+KVK,K~+I
+ (J+K+l)(J--K)~K,K’-ll,
Wa)
(J,) K,K~=K~K,K~.
(B3c)
The Hamiltonian
matrix is thus given as
(JaJp)~,~~pa,p(Qj)
GB
~K,~,,~=~K,K~$~~+~
+ (J)K,K~
[6jgp C
ZQj) :ZQjp)].
;
X (pZ>j,j, 2
I
gcQj)
(B4)
where the f; tensor has been symmetrized as in Sec. II A by
taking the average of the equivalent orderings ?i*p and
F ?i. The first term in Eq. (B 4) is the J=O Hamiltonian of
Eq. (3.7)) where cQj> is the grid introduced in Sec. II A.
The first term in curly brackets is the rotational Hamiltonian for an (asymmetric) molecule at a geometry {Qj}
and the second term is the rotation-vibration
coupling.
Equation (B4) shows that the introduction of a DVR for
the F vibrational-like degrees of freedom makes it relatively simple to apply the compZete Watson Hamiltonian to
a general chemical reaction without resort to approximations.
APPENDIX c: RADAU COORDINATES
Figure 10 shows the Radau coordinates28 for the special case of a three-atom system. We denoted by D the
center of mass of atoms A and C and by 0 the canonical
point satisfying the condition
&,=RB-DXL,.--D,
(Cl)
J. Chem. Phys., Vol. 97, No. 4, 15 August 1992
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2514
T. Seideman and W. H. Miller: Quantum-mechanical
i.e., 0 is the geometric mean of the distances from D to
atom B and to the triatomic center of mass, denoted by
c.m.
The total kinetic energy (for the rotational state) is
given in terms of Radau coordinates as36(b)
T=-za2
#
2ml @ -?&
xg
(l-22)
a2 #
w-2
cc21
-&
where R,= IRal, CZ= 1, 3, and z=cos y=R,*R3/R1R3.
For a more complete discussion of the coordinate system,
see Ref. 36.
The discrete variable representation of Eq. (C2) is
written as
Tj,jt=Tf’
.S.
4. Jr’r++T’f3,S. JIJI.tS.Jr’.,+
r
‘33
J3,J;
JpJ;
,+-&
&
1 1.1,
T?J& .,S.JlJld. J3J3’
+
3
(C3)
3sJ3
where ml and m3 are the
spectively. We used the
(3.12), and (3.14~) for
Gauss-Legendre DVR ‘4(f)
masses of atoms A and C, reequally spaced DVR of Eqs.
the radial coordinates and a
for the angular motion
Nb- 1
TFdt= lzo (j,lOW+l)Vljj)
Y
N,,--l
(C4)
where pi(z) are normalized Legendre polynomials
The absorbing potential was found most efficient when
placed in terms of the coordinate
Qr(WW)
(C5)
= I RI --R3 1,
i.e., l?(q) =I’[Qr(R1,R3,y)]
dence of Eq. (4.2).
with the functional
depen-
‘W. H. Miller, J. Chem. Phys. 62, 1899 (1975).
‘T. Yamamoto, J. Chem. Phys. 33, 281 (1960).
3(a) W. H. Miller, J. Chem. Phys. 61, 1823 (1974); (b) W. H. Miller,
S. D. Schwartz, and I. W. Tromp, ibid. 79, 4889 (1983).
4(a) R. Jaquet and W. H. Miller, J. Phys. Chem. 89, 2139 (1985); (b)
K. Yamashita and W. H. Miller, J. Chem. Phys. 82, 5475 (1985); (c)
J. W. Tromp and W. H. Miller, J. Phys. Chem. 90,3482 (1986); (d) J.
W. Tromp and W. H. Miller, Faraday Discuss. Chem. Sot. 84, 441
(1987).
‘R. E. Wyatt, Chem. Phys. Lett. 121, 301 (1985).
6(a) T. P. Park and J. C. Light, J. Chem. Phys. 85,587O (1986); (b) 88,
4897 (1988); (c) 91, 974 (1989); (d) M. Founargiotakis and J. C.
reaction probabilities
Light, ibid. 93, 633 (1990); (e) T. P. Park and J. C. Light, ibid. 94,
2946 (1991). (f) (in press).
7(a) G. Wahnstrom and H. Metiu, Chem. Phys. Lett. 134, 531 (1987);
J. Phys. Chem. 92, 3240 (1988); (b) G. Wahnstrom, B. Carmeli, and
H. Metiu, J. Chem. Phys. 88, 2478 (1988); (c) A. E. Depristo, K.
Haug, and H. Metiu, Chem. Phys. Lett. 155, 376 (1989).
’ (a) R. Lefebvre and N. Moiseyev, J. Chem. Phys. 93,7173 (1990); (b)
N. Ron, N. Moiseyev, and R. Lefebvre, J. Chem. Phys. 96, 8307
(1992).
‘P. N. Day and D. G. Truhlar, J. Chem. Phys. 94, 2045 (1991).
“D Thirumalai, B. C. Garrett, and B. J. Beme, J. Chem. Phys. 83,2972
(i985).
“C. W. McCurdy and B. C. Garrett, J. Chem. Phys. 84, 2630 (1986).
“T. Seideman and W. H. Miller, J. Chem. Phys. 95, 1768 (1991).
13T. Seideman and W. H. Miller, J. Chem. Phys. 96, 4412 (1992).
I4 (a) J. V. Lill, G. A. Parker, and J. C. Light, Chem. Phys. Lett. 89,483
(1982); (b) J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys.
82, 1400 (1985); (c) J. V. Lill, G. A. Parker, and J. C. Light, ibid. 85,
900 (1986); (d) 2. Bacic and J. C. Light, ibid. 85, 4594 (1986); 86,
3065 (1987); (e) R. M. Whitnell and J. C. Light, ibid. 90, 1774 (1989);
(f) S. E. Choi and J. C. Light, ibid. 92, 2129 (1990).
“D. T. Colbert and W. H. Miller, J. Chem. Phys. 96, 1982 (1992).
16(a) A. Goldberg and B. W. Shore, J. Phys. B 11, 3339 (1978); (b) C.
Leforestier and R. E. Wyatt, J. Chem. Phys. 78, 2334 (1983).
I7 (a) C. Cejan, D. Kosloff, R. Kosloff, and T. Reshef, Geophysics 50,
705 (1985); (b) R. Kosloff and D. Kosloff, J. Comput. Phys. 63, 363
(1986); (c) D. Neuhauser and M. Baer, J. Chem. Phys. 90, 4351
(1989); (d) 91,465l (1989); (e) D. Neuhauser, M. Baer, R. S. Judson,
and D. J. Kouri, ibid. 93, 312 (1990); (f) M. Baer, C. Y. Ng, and D.
Neuhauser, Chem. Phys. Lett. 169, 534 (1990); (g) D. Neuhauser, J.
Chem. Phys. 95, 4927 (1991); (h) I. Last and M. Baer, Chem. Phys.
Lett. 189, 84 (1992); I. Last, D. Neuhauser, and M. Baer, J. Chem.
Phys. 96, 2017 (1992).
‘*Other pointwise approaches are the pseudospectral method suggested
by Friesner [Chem. Phys. Lett. 116,39 (1985); J. Chem. Phys. 85, 1462
(1987)], and the colocation method of Yang and Peet [Chem. Phys.
L&t. 153, 93 (1988); J. Chem. Phys. 90, 1746 (1989)].
“W. H. Miller and B. M. D. D. Jansen op de Haar, J. Chem. Phys. 86,
6213 (1987).
*‘B. Simon, Phys. Lett. A 71, 211 (1979).
‘lU. Peskin and N. Moiseyev, J. Chem. Phys. 96, 2347 (1992).’
“J. K. G. Watson, Mol. Phys. 15, 479 (1968).
zr J. K. G. Watson, Mol. Phys. 19, 465 (1970).
24See, e.g., R. G. Newton, Scattering Theory of Waves and Particles
(Springer, Berlin, 1982).
25See, e.g. (a) I. M. Mills, in Molecular Spectroscopy: Modern Research,
edited by K. Narahari Rao and C. W. Nathens (Academic, New York,
1972), Vol. 1, p. 115, (b) M. R. Aliev and J. K. G. Watson, J. Mol.
Spectrosc. 3, 1 (1988).
26E B Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations
(McGraw-Hill, New York, 1955).
*‘J. C. Light (private communication).
*‘R. Radau, Ann. Sci. Cole Normale Superior 5, 311 (1868).
*‘A. J. C. Varandas, F. B. Brown, C. A. Mead, D. G. Truhlar, and N. C.
Blais, J. Chem. Phys. 86, 6258 (1987).
“D. C. Chatfield, D. G. Truhlar, and D. W. Schwenke, J. Chem. Phys.
94, 2040 (1991). (The solid circles in Fig. 4 were measured from Fig.
3 of Ref. 29 and are therefore approximate.)
3’J. Z. H. Zhang (private communication).
32P. Siegbahn and B. Liu, J. Chem. Phys. 68,2457 (1978); D. G. Truhlar
and C. J. Horowitz, ibid. 68, 2566 (1978); 71, 1514(E) (1979).
33J. M. Bowman, Adv. Chem. Phys. 61, 115 (1985).
34M Abramowitz and I. A. Stegun, Handbook of Mathematical Functions
(Dover, New York, 1965).
35J. H. Van Vleck, Rev. Mod. Phys. 23, 213 (1951).
36(a) F. T. Smith, Phys. Rev. L&t. 45, 1157 (1980); (b) B. R. Johnson
and W. P. Reinhardt, J. Chem. Phys. 85, 4538 (1986).
J. Chem. Phys., Vol. 97, No. 4, 15 August 1992
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